Cardiovascular fitness is commonly measured using $\mathrm{VO}_2$ max, the maximum amount of oxyge

Question

Cardiovascular fitness is commonly measured using $\mathrm{VO}_2$ max, the maximum amount of oxygen the body can use during intense exercise. The higher the value of $\mathrm{VO}_2$ max, the fitter the person is. In a study on cardiovascular fitness, the resting heart rate $x$, measured in beats/min and the $\mathrm{VO}_2$ max $y$, measured in ml/kg/min, of $8$ randomly selected people are given in the table below.

$x$	$78$	$75$	$65$	$64$	$62$	$56$	$53$	$48$
$y$	$36$	$41$	$47$	$48$	$48$	$56$	$58$	$63$

Draw a scatter diagram for these values, labelling the axes.[1]
Find the product moment correlation coefficient between $y$ and $x$ and comment on its value in the context of the data.[2]
Use an appropriate regression line to estimate the $\mathrm{VO}_2$ max of a person with a resting heart rate of $60$ beats/min, leaving your answer to the nearest integer. Comment on the reliability of your estimate.[3]
Use a specific example to explain why extrapolating using your regression line in part (iii) to predict the $\mathrm{VO}_2$ max of people with very high or very low resting heart rate is not advisable.[1]

Due to an instrumentation issue, all the resting heart rate values are to be increased by $v$ beats/min.

Find the value of the product moment correlation coefficient, and the gradient and $y$-intercept of the regression line of $y$ on $x$ after these adjustments, leaving your answer in terms of $v$ where applicable.[3]

Accepted Answer

(i) Scatter shown. (ii) $r\approx-0.991$, strong negative linear correlation. (iii) $52$ ml/kg/min; interpolation with a strong linear fit. (iv) At 130 beats/min the model predicts an impossible negative value. (v) $r$ and slope unchanged; intercept $103.3947852\ldots+0.8585993649\ldots v$.

2025 MI P2 Q11

2025 MI P2 Q11

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